Question

How do you find `lim (x-1)/(x-2sqrtx+1)` as `x->1^+`?

Answer 1

See below.

Explanation:

`(x-1)/(x-2sqrtx+1) = ((sqrtx+1)(sqrtx-1))/(sqrtx-1)^2`

`= (sqrtx+1)/(sqrtx-1)` `" "` for `x != 1`

`lim_(xrarr1^+) (x-1)/(x-2sqrtx+1) = lim_(xrarr1^+) (sqrtx+1)/(sqrtx-1)`

`= 2/0^+ = oo`

The notation in the last line is intended to indicate that the numerator approaches `2` and the denominator goes to `0` through positive values.
The result is a quotient that is increasing without bound.